Mean Value On Some Arithmetical Functions

Arithmetical function is a real-valued or complex-valued function defined on the set of positive integers. The research on the mean value properties of arithmetical functions is an important subject. It is well known that the values of many important arithmeti-cal functions are irregular. Therefore, we usually grasp the rules of these functions by studying their mean value properties.The famous Fibonacci series has always attracted lots of attention because of its simple regulation and rich content. Fibonacci series have direct application in modern physics, quasi-crystalline structure and other fields. American Mathematical Society has published "Fibonacci series" quarterly since 1963, which is devoted to research in this area.In 1993, American-Romanian number theorist Florentin Smarandache published a book named "Only Problems, Not Solutions!". In this book, he proposed 105 unsolved problems about sequences and arithmetical functions, Many scholars have studied these problems, and obtained some important results unceasingly.In this dissertation, we studied the Fibonacci series and two arithmetical functions proposed by Florentin Smarandache, and got some mean value asymptotic formulas about them. Specifically, the main achievements in this paper are as follows:1. We studied the digital sum function in the binary, and got its m-power mean value properties. Then we studied the digital sum function in the p-nary, and got its m-power mean value properties.2. Using conjecture, induction and recursion methods, we studied a counting function relating to the Fibonacci series, and gave its r-power mean value asymptotic formula.3. Using the elementary methods and analytic methods, we studied the hybrid mean value involving the square root sequence functionα2(n) and the Euler totient functionφ(n), and obtained two asymptotic formulas.